\[ \newcommand{\test}{The Wikibook about \LaTeX} \newcommand{\manugood}{{}_👚} \newcommand{\agrigood}{{}_🌽} \] Can't specialize in manufactured goods. \[ 👚 = z_{\manugood} N_{\manugood} + \gamma z_{\manugood} + z_{\manugood} \gamma + z_👚 \gamma +\]

Autarky

Assumptions:

Two goods: manufactured goods and \(👚\) agricultural goods \(🌽\)
Consumer needs to have non-zero food, but can have zero manufactured goods and still survive.
Both goods are produced with CRS technology.
No leisure decisions, I guess, just to keep things easy. Set price of 🌽 as numeraire?

Formal Definition of Equilibrium:

Click to expand definition

Consumer's Problem:

Taking prices \(\{w,p\}\) as given, chooses \(\{👚,🌽\}\) to solve \[\max_{👚,🌽} \ln (🌽) + \gamma \ln (👚+1)\] subject to the constraints that \[👚,🌽 \geq 0\] \[p👚 + 🌽 \leq w + \pi_👚 + \pi_🌽\]

Manufacturing Firm's Problem:

Taking prices \(\{w,p\}\) as given, chooses \(\{N_👚\}\) to solve \[\max p z_👚 N_👚 - w N_👚\]

Agricultural Firm's Problem:

Taking prices \(\{w,p\}\) as given, chooses \(\{N_🌽\}\) to solve \[\max z_🌽 N_🌽 - w N_🌽\]

Market Clearing:

\[N_👚 + N_🌽 = 1\] \[ 👚 = z_👚 N_👚 \] \[ 🌽 = z_🌽 N_🌽 \]

Solution to firms' problems:

For non-zero finite output, it must be that \[p z_👚 = w = z_🌽\] and for zero output of 👚, it must be that \[w \geq p z_👚\] In either case, profits will be zero.

Solution to consumer's problem:

In an interior solution where \(👚>0\), \[\frac{(👚+1)}{\gamma 🌽} = p\] \[ p👚 + 🌽 = w \] \[🌽 = \frac{w+p}{\left(1+\gamma\right)}\] \[👚 = \frac{w\gamma-p}{p\left(1+\gamma\right)}\] And in a solution where \(👚=0\), \[🌽 = w\] Note that interior solution requires that \(p < w\gamma\)

Click to expand work for consumer problem

Assume profits are zero. Will be true in equilibrium, so no big deal. Then Assuming interior solution where \(🌽,👚>0\), Lagrangian is: \[\mathcal{L} = \ln (🌽) + \gamma \ln (👚+1) - \lambda [ p👚 + 🌽 - w ] \] FOCs: \[{1 \over 🌽} = \lambda\] \[{\gamma \over (👚+1)} = \lambda p\] \[p👚 + 🌽 = w \] Implies \[{\gamma \over (👚+1)} = {1 \over 🌽} p\] Solve this to express each variable in terms of the other: \[🌽=👚\frac{p}{\gamma}+\frac{p}{\gamma}\] \[👚=🌽\frac{\gamma}{p}-1\] Combine these with budget constraint to solve explicitly for allocation variables: \[🌽 = \frac{w+p}{\left(1+\gamma\right)}\] \[👚 = \frac{w\gamma-p}{p\left(1+\gamma\right)}\]

Combine problems to get equilibrium

If \(\gamma z_👚 > 1\), then we have the interior solution: \[🌽 = z_🌽 \cdot \left( \frac{1+z_👚}{z_👚\left(1+\gamma\right)} \right) \] \[👚 = \frac{\gamma z_👚 - 1}{\left(1+\gamma\right)}\] \[N_🌽 = \frac{🌽}{z_{🌽}} = \frac{\left(z_{👚}+1\right)}{z_{👚}\left(1+\gamma\right)}\] \[N_{👚} = \frac{👚}{z_{👚}}=\frac{\gamma-\frac{1}{z_{👚}}}{1+\gamma}\] Otherwise, if \(\gamma z_👚 \leq 1\), \[N_🌽 = 1\] \[N_👚 = 0\] \[🌽 = w = z_🌽\]

Click to expand work for solving for the equilibrium allocations

For an interior solution, we have the following characterizing equations:
From consumer: \[\frac{(👚+1)}{\gamma 🌽} = p\] \[ p👚 + 🌽 = w \] \[🌽 = \frac{w+p}{\left(1+\gamma\right)}\] \[👚 = \frac{w\gamma-p}{p\left(1+\gamma\right)}\] \[🌽,👚>0\] From firms: \[p z_👚 = w = z_🌽\] From Market Clearing \[N_👚 + N_🌽 = 1\] \[ Y_👚 = z_👚 N_👚 \] \[ Y_🌽 = z_🌽 N_🌽 \]
Note how this implies that \[p=\frac{ z_🌽 }{ z_👚 }\] And substitute out the prices to get allocations: \[🌽 = \frac{ z_🌽 + {z_🌽 \over z_👚} }{\left(1+\gamma\right)} = z_🌽 \frac{1+z_👚}{z_👚\left(1+\gamma\right)} \] \[👚 = \frac{z_🌽 \gamma-{z_🌽 \over z_👚}}{{z_🌽 \over z_👚}\left(1+\gamma\right)} = \frac{\gamma z_👚 - 1}{\left(1+\gamma\right)}\] \[N_🌽 = \frac{🌽}{z_{🌽}} = \frac{\left(z_{👚}+1\right)}{z_{👚}\left(1+\gamma\right)}\] \[N_{👚} = \frac{👚}{z_{👚}}=\frac{\gamma-\frac{1}{z_{👚}}}{1+\gamma}\] Note that the above equations are consistent with \(👚>0\) iff \(\gamma z_👚 > 1\). If this isn't the case, then it must instead be that \(👚=0\) and \[N_🌽 = 1\] \[N_👚 = 0\] \[🌽 = w = z_🌽\]